Found problems: 85335
2019 USA TSTST, 1
Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$,
[list]
[*] the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and
[*] if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$.
[/list]
[i]Evan Chen[/i]
2019 Romanian Master of Mathematics Shortlist, A2
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
$$
[i](B. Serankou, M. Karpuk)[/i]
2021 AMC 12/AHSME Spring, 8
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines?
$\textbf{(A)}\ 5\frac{1}{2} \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 6\frac{1}{2} \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 7\frac{1}{2}$
2018 MIG, 6
How many more hours are in $10$ years than seconds in $1$ day?
$\textbf{(A) }1000\qquad\textbf{(B) }1100\qquad\textbf{(C) }1150\qquad\textbf{(D) }1200\qquad\textbf{(E) }1300$
2022 New Zealand MO, 1
$ABCD$ is a rectangle with side lengths $AB = CD = 1$ and $BC = DA = 2$. Let $ M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\angle PCB$.
2012 Iran MO (3rd Round), 1
$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$.
[i]Proposed by Mohammad Gharakhani[/i]
2022 Durer Math Competition (First Round), 3
Paraflea makes jumps on the plane, starting from the origin $(0, 0)$. From point $(x, y)$ it may jump to another point of the form $(x + p, y + p^2)$, where $p$ is any positive real number. (The value of $p$ may differ for each jump.)
a) Is there any point in quadrant $I$ which cannot be reached by the flea? (Quadrant $I$ contains points $(x, y)$ for which $x$ and $y$ are positive real numbers.)
b) What is the minimum number of jumps that the flea must make from the origin so that it gets to the point $(100, 1)$?
2018 All-Russian Olympiad, 5
On the circle, 99 points are marked, dividing this circle into 99 equal arcs. Petya and Vasya play the game, taking turns. Petya goes first; on his first move, he paints in red or blue any marked point. Then each player can paint on his own turn, in red or blue, any uncolored marked point adjacent to the already painted one. Vasya wins, if after painting all points there is an equilateral triangle, all three vertices of which are colored in the same color. Could Petya prevent him?
2022 Belarus - Iran Friendly Competition, 2
Let $P(x)$ be a polynomial with rational coefficients such that $P(n)$ is integer for all
integers $n$. Moreover: $gcd(P(1), \ldots , P(k), \ldots) = 1$. Prove that every integer $k$ can be represented
in infinitely many ways of the form
$\pm P(1) \pm P(2) \pm \ldots \pm P(m)$,
for some positive integer $m$ and certain choices of $\pm$.
2004 AMC 12/AHSME, 12
Let $ A \equal{} (0,9)$ and $ B \equal{} (0,12)$. Points $ A'$ and $ B'$ are on the line $ y \equal{} x$, and $ \overline{AA'}$ and $ \overline{BB'}$ intersect at $ C \equal{} (2,8)$. What is the length of $ \overline{A'B'}$?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 2 \plus{} \sqrt 2\qquad \textbf{(E)}\ 3\sqrt 2$
2009 Tournament Of Towns, 4
Three planes dissect a parallelepiped into eight hexahedrons such that all of their faces are quadrilaterals (each plane intersects two corresponding pairs of opposite faces of the parallelepiped and does not intersect the remaining two faces). One of the hexahedrons has a circumscribed sphere. Prove that each of these hexahedrons has a circumscribed sphere.
2007 Indonesia TST, 3
Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.
2004 VTRMC, Problem 5
Let $f(x)=\int^x_0\sin(t^2-t+x)dt$. Compute $f''(x)+f(x)$ and deduce that $f^{(12)}(0)+f^{(10)}(0)=0$.
2019 Kyiv Mathematical Festival, 1
A bunch of lilac consists of flowers with 4 or 5 petals. The number of flowers and the total number of petals are perfect squares. Can the number of flowers with 4 petals be divisible by the number of flowers with 5 petals?
2005 Irish Math Olympiad, 1
Let $ X$ be a point on the side $ AB$ of a triangle $ ABC$, different from $ A$ and $ B$. Let $ P$ and $ Q$ be the incenters of the triangles $ ACX$ and $ BCX$ respectively, and let $ M$ be the midpoint of $ PQ$. Prove that: $ MC>MX$.
2017 NZMOC Camp Selection Problems, 5
Find all pairs $(m, n)$ of positive integers such that the $m \times n$ grid contains exactly $225$ rectangles whose side lengths are odd and whose edges lie on the lines of the grid.
2018 Puerto Rico Team Selection Test, 3
Let $A$ be a set of $m$ positive integers where $m\ge 1$. Show that there exists a nonempty subset $B$ of $A$ such that the sum of all the elements of $B$ is divisible by $m$.
Russian TST 2022, P1
Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities
$$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$
Prove that the degrees of the three polynomials are all even.
2010 Switzerland - Final Round, 3
For $ n\in\mathbb{N}$, determine the number of natural solutions $ (a,b)$ such that
\[ (4a\minus{}b)(4b\minus{}a)\equal{}2010^n\]
holds.
2013 Online Math Open Problems, 8
How many ways are there to choose (not necessarily distinct) integers $a,b,c$ from the set $\{1,2,3,4\}$ such that $a^{(b^c)}$ is divisible by $4$?
[i]Ray Li[/i]
2010 Baltic Way, 8
In a club with $30$ members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of $10$ members such that no one in the group has received a hat from another one in the group.
2014 Brazil Team Selection Test, 4
Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation
\[ f(f(f(n))) = f(n+1 ) +1 \]
for all $ n\in \mathbb{Z}_{\ge 0}$.
2003 Czech-Polish-Slovak Match, 2
In an acute-angled triangle $ABC$ the angle at $B$ is greater than $45^\circ$. Points $D,E, F$ are the feet of the altitudes from $A,B,C$ respectively, and $K$ is the point on segment $AF$ such that $\angle DKF = \angle KEF$.
(a) Show that such a point $K$ always exists.
(b) Prove that $KD^2 = FD^2 + AF \cdot BF$.
2010 Today's Calculation Of Integral, 603
Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$.
Please solve the problem without using partial differentiation for those who don't learn it.
1961 Waseda University entrance exam/Science and Technology
2014 BMO TST, 1
Prove that for $n\ge 2$ the following inequality holds:
$$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$